Question

Suppose Z follows a standard normal distribution (Z ∼ N(0, 1)), and you're tasked with finding various probabilities associated with Z.

(a) What is the probability that Z is less than 0.5, i.e., P(Z < 0.5)?

(b) Determine the probability of Z being exactly equal to 0.5, i.e., P(Z = 0.5).

(c) Calculate the probability that Z is greater than or equal to 2.3, i.e., P(Z ≥ 2.3).

(d) Find the probability that Z falls within the interval [-1.4, 0.6], i.e., P(-1.4 ≤ Z ≤ 0.6).

(e) Identify the value of z0 such that the absolute value of Z falls within ∣Z∣ ≤ z0 with a probability of 0.32, i.e., P(∣Z∣ ≤ z0) = 0.32.

Answer 1

To find probabilities associated with a standard normal distribution: (a) Calculate the area under the curve to the left of 0.5. (b) Since the probability of a single point in a continuous distribution is 0, the probability of Z being exactly 0.5 is 0. (c) Calculate the area under the curve to the right of 2.3. (d) Calculate the difference between the areas to the left of -1.4 and 0.6. (e) Find the z-score that corresponds to the 0.16th percentile.

Step-by-step explanation:

(a) To find the probability that Z is less than 0.5, we need to calculate the area under the standard normal curve to the left of 0.5. Using a z-table or calculator, the area is approximately 0.6915.

(b) Since the standard normal distribution is continuous, the probability of Z being exactly equal to 0.5 is 0. You cannot have a single point probability in a continuous distribution.

(c) To find the probability that Z is greater than or equal to 2.3, we need to calculate the area under the standard normal curve to the right of 2.3. Using a z-table or calculator, the area is approximately 0.0107.

(d) To find the probability that Z falls within the interval [-1.4, 0.6], we need to calculate the area under the standard normal curve between -1.4 and 0.6. This is equivalent to finding the difference between the areas to the left of -1.4 and 0.6. Using a z-table or calculator, the area is approximately 0.7745.

(e) To find the value of z0 such that the absolute value of Z falls within ∣Z∣ ≤ z0 with a probability of 0.32, we need to find the z-score that corresponds to the 0.16th percentile (half of 0.32). Using a z-table or calculator, the z-score is approximately -0.507.